YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(x0) → b(b(x0))
b(b(b(x0))) → a(x0)

Proof

1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 0
0 1
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 1
1 2
· x1 +
-∞ -∞
-∞ -∞
the rule
a(x0) → b(b(x0))
remains.

1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
0 0 0
0 0 1
0 1 1
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[a(x1)] =
1 2 2
2 3 3
2 3 3
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
all rules could be removed.

1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.